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Abstract

We consider a semilinear fourth-order elliptic equation with a right-hand side nonlinearity
which exhibits an asymmetric growth at +∞ and at −∞. Namely, it is linear at −∞ and
superlinear at +∞. Combining variational methods with Morse theory, we show that the
problem has at least two nontrivial solutions, one of which is negative.

Keywords:

1 Introduction

where is the biharmonic operator, and Ω is a bounded smooth domain in (), and the first eigenvalue of −△ in .

The conditions imposed on are as follows:

(H1) , for all and for all and all ;

(H2) there exist and such that for all , and , where , if ;

(H3) uniformly for , where l is a nonnegative constant;

(H4) there exist such that for , we have

(2)

(3)

and

(4)

(H5) there exist and an integer such that

(5)

the first and the last inequality are strict on sets (not necessary the same) of
positive measure, and

(6)

In view of the conditions (H3) and equation (3) in (H4), it is clear that for all , is linear at −∞ and superlinear at +∞. Clearly, is a trivial solution of problem (1). It follows from (H1) and (H2) that the functional

(7)

is on the space with the norm

where . Under the condition (H2), the critical points of I are solutions of problem (1). Let be the eigenvalues of and be the eigenfunction corresponding to . In fact, . Let denote the eigenspace associated with . Throughout this article, we denoted by the norm and . The aim of this paper is to prove a multiplicity theorem for problem (1) when the
nonlinearity term exhibits an asymmetric behavior as approaches +∞ and −∞. In the past, some authors studied the following elliptic problem:

(8)

with asymmetric nonlinearities by using the Fučík spectrum of the operator . This approach requires that exhibits linear growth at both +∞ and −∞ and that the limits exist and belong to ℝ. See the works of Các [1], Dancer and Zhang [2], Magalhães [3], de Paiva [4], Schechter [5] and the references therein. Equations with nonlinearities which are superlinear in
one direction and linear in the other were investigated by Arcoya and Villegas [6] and Perera [7]. They let the nonlinearity be line at −∞ and satisfy the Ambrosetti-Rabinowitz condition at +∞. Particularly,
it is worth noticing paper [8]. The authors relax several of the above restrictions on the nonlinearity . Their nonlinearity is only measurable in . The limit as of need not exist and the growth at −∞ can be linear or sublinear. Furthermore, their
nonlinearity does not satisfy the famous AR-condition. They use the truncated skill of first order
weak derivative to verify the (PS) condition and obtain multiple solutions for problem
(1) by combining variational methods and Morse theory.

To the authors’ knowledge, there seem to be few results on problem (1) when is asymmetric nonlinearity at positive infinity and at negative infinity. However,
the method in [8] cannot be applied directly to the biharmonic problems. For example, for the Laplacian
problem, implies , where , . We can use or as a test function, which is helpful in proving a solution nonnegative. While for
the biharmonic problems, this trick fails completely since does not imply (see [[9], Remark 2.1.10] and [10,11]). As far as this point is concerned, we will make use of the new methods to overcome
it.

This fourth-order semilinear elliptic problem can be considered as an analogue of
a class of second-order problems which have been studied by many authors. In [12], there was a survey of results obtained in this direction. In [13], Micheletti and Pistoia showed that admits at least two solutions by a variation of linking if is sublinear. Chipot [14] proved that the problem has at least three solutions by a variational reduction method and a degree argument.
In [15], Zhang and Li showed that admits at least two nontrivial solutions by Morse theory and local linking if is superlinear and subcritical on u.

In this article, under the guidance of [8], we consider multiple solutions of problem (1) with the asymmetric nonlinearity by
using variational methods and Morse theory.

where is a positive constant and is a sequence which converges to zero. By a standard argument, in order to prove
that has a convergence subsequence, we have to show that it is a bounded sequence. To
do this, we argue by contradiction assuming that for a subsequence, denoted by , we have

Without loss of generality we can assume for all and define . Obviously, and then it is possible to extract a subsequence (denoted also by ) such that

(11)

(12)

(13)

(14)

where and . Dividing both sides of inequality (10) by , we obtain

Passing to the limit we deduce from equation (11) that

(15)

for all .

Now we claim that a.e. . To verify this, let us observe that by choosing in equation (15) we have

(16)

where . But, on the other hand, from (H3) and equation (3) in (H4), we have

for some positive constant . Moreover, using a.e. , equation (13) and the superlinearity of f, we also deduce

Therefore, if we will obtain by Fatou’s lemma that

which contradicts inequality (16). Thus and the claim is proved.

Clearly, , by (H3), there exists such that for a.e. . By using Lebesgue dominated convergence theorem in equation (15), we have

(17)

for all . This contradicts . □

Lemma 2.3Let, where. Ifis an integer, , a.e. on Ω and the inequality is strict on a set of positive measure, then there existssuch that

for all.

Proof We claim that there exists a constant such that

(18)

for all . In fact, if not, there exists a sequence such that

for all , which implies for all n. By the homogeneity of the above inequality, we may assume that and

(19)

for all n. It follows from the weak compactness of the unit ball of W that there exists a subsequence, say , such that weakly converges to u in W. Now Sobolev’s embedding theorem suggests that converges to u in . From inequality (19) we obtain

Moreover one has

Hence we have

and

which implies that and on a positive measure subset. It contradicts the unique continuation property of
the eigenfunction. □

3 Computation of the critical groups

It is well known that critical groups and Morse theory are the main tools in solving
elliptic partial differential equation. Let us recall some results which will be used
later. We refer the readers to the book [16] for more information on Morse theory.

Let H be a Hilbert space and be a functional satisfying the (PS) condition or (C) condition, and be the qth singular relative homology group with integer coefficients. Let be an isolated critical point of I with , , and U be a neighborhood of . The group

is said to be the qth critical group of I at , where .

Let be the set of critical points of I and , the critical groups of I at infinity are formally defined by [17]

From the deformation theorem, we see that the above definition is independent of
the particular choice of . If then

(20)

For the convenience of our proof, we first recall two interesting results and prove
two important propositions.

Proof Under the guidance of [8] and [18], we begin to prove this result. Let . Indeed, it follows from above Proposition 3.1 that I and have same critical set. Since is dense in E, invoking Proposition 16 of Palais [20], we have

(21)

From equations (20) and (21), we see that in order to prove the proposition, it suffices
to show that

(22)

In order to prove equation (22), we proceed as follows. We define the sets

and

Consider the map defined by

Clearly, is a continuous homotopy and for all . Therefore, is contractible in itself.

By equation (3) in (H4), given any , we can find such that

(23)

Similarly, from condition (H3), and by choosing even bigger if necessary, we observe that there is a number such that

(24)

Moreover, by condition (H2), we have

(25)

for some .

Let . By inequalities (23), (24), and (25), for all we have

(26)

Recalling that is arbitrary, from (26), we have

(27)

Using formula (4) in condition (H4), we see that there exist constants and such that

(28)

By (H2) and formula (2) in condition (H4), we have

(29)

for some . By inequalities (28) and (29), for any we have

(30)

where C is a positive constant. Let be the continuous embedding map. Let denote the duality brackets for the pair . We let , and so

(31)

Then, from equation (27), we obtain

(32)

From conditions (H2) and (H3), we see that given , we can find such that

(33)

Using inequality (33), we have

for , where is defined as

and is a positive constant. So is coercive, thus we find such that . We pick

Then inequality (32) implies that we can find such that

Moreover, the implicit function theorem implies that .

By the choice of a, we have

(34)

We define the set . The map defined by

(35)

is a continuous deformation of , and for all (see equations (34) and (35)). Therefore, is a strong deformation retract of . Hence we have

(36)

Recalling that in the first part of the proof, we established that is contractible. This yields

Combining with equation (36) leads to equation (22), which completes the proof. □

Proposition 3.4If the assumptions of Theorem 2.1 hold, then

where (Vbeing defined in Lemma 2.3).

Proof By condition (H5), given , we can find such that

(37)

Since V is finite dimensional, all norms are equivalent. Thus we can find small such that

(38)

for all . Taking inequalities (37) and (38) into account, for all with we have

(39)

Similar to the proof of Lemma 2.3, there exists such that

(40)

for all and .

On the other hand, for given , it follows from (H2) and (H5) that

(41)

for all and . By (41) and Lemma 2.3, we have

(42)

for all . From inequality (42), we infer that for ρ small enough we have

(43)

From inequalities (40) and (43), we know that I has a local linking at 0. Then invoking Proposition 2.3 of Bartsch and Li [17], we obtain . □

4 Proof of the main result

Proof of Theorem 2.1 We consider the following problem:

where

Define a functional by

where , then . Obviously, by conditions (H1) and (H3), we know that is coercive and boundedness from below. Thus we can find such that

(44)

Next, we claim that . By condition (H5), given , there exists such that

(45)

For s small enough, it follows from inequality (45) that

and thus, by equation (44), , so . From condition (H1) and strong maximum principle, we have and

Since is an interior point of , from Proposition 3.1, we know

(46)

Let , be such that . We consider the sublevel sets

Suppose that 0 and are the only critical points of I. Otherwise, we have a second nontrivial smooth solution and so we are done. By Proposition 3.3,
we have

(47)

We know that I satisfies the (PS) condition (see Lemma 2.2). Hence choosing small enough, we have

(48)

(see Proposition 3.4). Because of equations (47) and (48), using Proposition 3.2,
we obtain

If , then there is a critical point of I such that

If , then there is a critical point of I such that

(49)

Since , from equations (46) and (49), we see that . It is obvious that . Therefore and are two solutions of problem (1). □

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

The authors read and approved the final manuscript.

Acknowledgements

The authors would like to thank the referees for valuable comments and suggestions
in improving this article. This study was supported by the National NSF (Grant No. 11101319)
of China and Planned Projects for Postdoctoral Research Funds of Jiangsu Province
(Grant No. 1301038C).